Anubhav Math Week2 L2 Learning Sheet

The Mathematics of Water Safety L2 Math Learning Sheet - Week 2: Ratios, Proportions & Percentages In the style of Ian Stewart Mathematics in Real Life Mathematics isn't just numbers in textbooks. It's a powerful tool hiding everywhere around us. When Manthan found that his water calculations didn't predict who got sick, he discovered how mathematics helps solve real problems. The water safety mystery shows us that mathematics is like detective work - we find patterns, create rules, and use them to predict what will happen next. Understanding Ratios in Water Safety The Body Water Ratio When we calculate water needs using body weight × 35ml, we're using a special mathematical relationship called a ratio. What makes this ratio special? It works for everyone: small children, teenagers, adults A 20kg person needs 700ml, a 40kg person needs 1400ml The ratio stays the same: 35ml per kg Why ratios are powerful: They let us scale up or down while keeping relationships the same. This is why the same formula works for Rohit (30kg) and Meera (26kg). Village Contamination Ratios Chitra discovered that villages with more drains per person had cleaner water. This is another ratio relationship: Examples: Badegaon: 2500 people, 12 drains = 208 people per drain → Clean water Khapa: 2000 people, 15 drains = 133 people per drain → Dirty water The pattern: More drains per person = cleaner water. Fewer drains per person = dirtier water. Mathematical thinking: When you have the right ratio of infrastructure to people, systems work better. Proportions and the Contamination Multiplier Understanding Proportional Increases When water is contaminated, your body needs proportionally more water. This means the amount increases in a mathematical relationship: The Proportion Pattern: Clean water: 1.0 times normal amount (no extra) Dirty water: 1.2 times normal amount (20% extra) Very dirty water: 1.4 times normal amount (40% extra) Example: If Rohit normally needs 2000ml: With clean water: 2000ml × 1.0 = 2000ml With dirty water: 2000ml × 1.2 = 2400ml With very dirty water: 2000ml × 1.4 = 2800ml Why proportions matter: Your body's response to contamination isn't random - it follows mathematical rules we can predict and use. The Mathematics of Health Prediction Using proportions, we can predict who will get sick: Rohit's case: Needed: 2050ml × 1.4 = 2870ml (very dirty water) Actually drank: 2100ml Deficit: 2870 - 2100 = 770ml (27% short) Prediction: Will get sick ✓ Priya's case: Needed: 2230ml × 1.0 = 2230ml (clean water) Actually drank: 2300ml Surplus: 2300 - 2230 = 70ml (3% extra) Prediction: Will stay healthy ✓ Mathematical power: Numbers help us predict the future! Percentage Calculations for Problem Solving Understanding Percentage Deficits Percentages help us understand how serious a problem is: Calculation method: Find the difference: Needed - Actually consumed Divide by what was needed Multiply by 100 to get percentage Example - Arjun: Needed: 2244ml, Drank: 1900ml Deficit: 2244 - 1900 = 344ml Percentage: (344 ÷ 2244) × 100 = 15% deficit Pattern discovered: Athletes with more than 10% deficit got sick. Athletes with surpluses stayed healthy. Cost-Benefit Percentage Analysis Mathematics also helps with money decisions: Prevention costs: ₹60 per month per family Treatment costs: ₹1200 per month per family (when sick) Percentage savings: Difference: ₹1200 - ₹60 = ₹1140 saved Percentage: (₹1140 ÷ ₹1200) × 100 = 95% savings! Mathematical conclusion: Prevention saves 95% of costs compared to treatment. How Mathematical Thinking Solves Problems The Problem-Solving Process Observe patterns: Notice that some calculations don't match reality Ask mathematical questions: "What ratios or relationships might explain this?" Collect data: Gather numbers to test your mathematical ideas Find relationships: Look for ratios, proportions, or percentage patterns Create mathematical models: Make formulas that represent the real world Test predictions: Use your mathematics to predict new situations Apply solutions: Use mathematical insights to help others Why This Approach Works Mathematics gives us superpowers: Prediction: We can know what will happen before it happens Optimization: We can find the best solutions using numbers Scaling: We can apply solutions from one place to many places Precision: We can be exact instead of just guessing Real-World Applications of These Mathematical Principles Environmental Mathematics The same ratio and proportion thinking applies to: Air pollution: Calculating safe exposure ratios for different activities Soil contamination: Determining proportional treatment needs for different contamination levels Weather prediction: Using percentage changes to forecast environmental conditions Health Mathematics Nutrition planning: Calculating proportional vitamin and mineral needs Exercise science: Determining optimal work-to-rest ratios for different fitness levels Medicine dosing: Using body weight ratios to calculate safe medication amounts Economics and Planning Resource allocation: Using proportional thinking to distribute limited resources fairly Infrastructure planning: Calculating optimal ratios of services to population Risk assessment: Using percentage analysis to make informed decisions The Beauty of Mathematical Discovery What Makes Mathematics Beautiful? Universal patterns: The same mathematical principles appear in many different situations Predictive power: Mathematics lets us see into the future using present information Elegant solutions: Complex problems often have surprisingly simple mathematical solutions Creative problem-solving: Mathematics gives us tools to approach problems in new ways Your Mathematical Achievement When you discovered contamination multipliers and percentage thresholds, you engaged in authentic mathematical research. You: Identified mathematical patterns in real-world data Created mathematical models to represent complex systems Used mathematical prediction to solve practical problems Applied mathematical optimization to improve community health This is exactly what professional mathematicians do - they find patterns, create models, and use mathematical thinking to understand and improve the world. Questions for Further Mathematical Exploration Investigations You Could Pursue Seasonal mathematics: How do contamination ratios change during different weather patterns? Population scaling: How do water safety formulas change for cities vs. villages? Prevention optimization: What mathematical models could minimize both cost and health risk? Prediction accuracy: How could statistical methods improve the reliability of your formulas? Mathematical Skills to Develop Statistical analysis: Understanding averages, trends, and data reliability Algebraic modeling: Creating mathematical equations that represent real systems Geometric thinking: Visualizing mathematical relationships through graphs and diagrams Probabilistic reasoning: Understanding uncertainty and risk in mathematical predictions Connection to the Global Mathematical Community Your water safety mathematics connects you to an international community of researchers working on similar problems: Water engineers use the same proportional thinking to design treatment systems Public health mathematicians model disease prevention using similar ratio analysis Environmental scientists apply percentage-based analysis to predict ecological changes Policy makers use cost-benefit mathematics to make decisions affecting millions of people You are now part of this mathematical community - people who use quantitative reasoning to make the world better. "Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." - William Paul Thurston Acharya Guide - Teaching Notes Pre-Session Preparation: Mathematical concept review: Solid understanding of ratios, proportions, percentages Real data preparation: Gather additional examples of mathematical relationships from local context Calculation tools: Ensure calculators available, prepare ratio/proportion templates Extension materials: Advanced problems for students who grasp concepts quickly During Session (30-35 minutes): Opening (5 mins): "Mathematics helps us find hidden patterns and solve real problems" Core Mathematical Development (20-25 mins): Pattern discovery: Guide students to find mathematical relationships themselves Concept connections: Link ratios, proportions, percentages to story discoveries Real-world applications: Show how mathematical thinking applies beyond water safety Problem-solving practice: Use mathematical models to predict new scenarios Mathematical Reflection (5 mins): "How does mathematical thinking change how you see problems?" Key Teaching Points: Mathematics as tool: Emphasize practical problem-solving power of mathematical thinking Pattern recognition: Develop ability to see mathematical relationships in complex situations Model building: Help students understand how mathematical formulas represent reality Prediction power: Show how mathematics enables forecasting and planning Common Student Questions & Responses: "Why do ratios stay the same?" → "Because mathematical relationships reflect natural laws" "How do we know our formula is right?" → "We test it against reality and adjust if needed" "Where else can we use this thinking?" → "Everywhere! Look for patterns and relationships" Assessment Indicators: Students demonstrate understanding when they: Can explain why ratios work across different scales Use proportional reasoning to solve new water safety problems Calculate percentages to analyze and compare different scenarios Apply mathematical thinking to predict outcomes in unfamiliar situations Extension for Advanced Students: "Create a mathematical model for a different community health challenge using the same ratio and proportion principles" Focus: Develop mathematical thinking as a powerful tool for understanding and improving the real world.